Under very general assumptions, as in [1]}, one is usually able to see that the original problem is in duality with a problem of optimal control of the HJB equation. However, existence of solutions for this dual problem is much harder to come by and requires particular structural conditions. Essentially, the dual problem has a solution if the Hamilton-Jacobi-Bellman equation admits a regular solution. This is of course rather difficult to obtain. Regularity results for the Hamilton-Jacobi-Bellman equation where the control appears in the volatility as in Fleming and Soner [2]} Chapter \(IV.4\) , usually rely upon three things : the regularity of the coefficients of the diffusion and of the costs functionals, the compactness of the control set and finally the uniform parabolicity of the equation. The last point means that there must be some \(\Lambda ^- >0\)  such that the volatility coefficient satisfies (uniformly on the time/state/control variables) : \(\sigma ^t\sigma \ge \Lambda ^- I_d\) .
